Characterization of viscoelastic behavior of basalt fiber asphalt mixtures based on discrete and continuous spectrum models

In order to analyze the differences between the master curves of relaxation modulus E(t) and creep compliance J(t) obtained from discrete and continuous spectrum models, and to comprehensively evaluate the effect of basalt fiber content on the viscoelastic behavior of asphalt mixtures, complex modulus tests were conducted for asphalt mixtures with fiber content of 0%, 0.1%, 0.2% and 0.3%, respectively. Consequently, the master curves of Viscoelastic Parameters of asphalt mixtures were constructed according to the generalized Sigmoidal model(GSM) and the approximate Kramers-Kronig (K-K) relationship. Then, transformation of master curves using discrete and continuous spectrum models to obtain the models of E(t) and J(t) containing all viscoelastic information. Also, the accuracy of the models of E(t) and J(t) was evaluated. The results show that the addition of basalt fibers improves the strength, stress relaxation and deformation resistance of asphalt mixtures. It is worth noting that basalt fibers achieve the improvement of asphalt mixtures by changing their internal structure. Considering the different viscoelastic master curves at four dosages, the optimum fiber dosage was 0.2%. In addition, both discrete and continuous model conversion methods can obtain high accuracy conversion results.

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Introduction
The viscoelastic behavior of asphalt mixture directly affects the road performance of asphalt

  Et and  
Jt in the time domain.Finally, the accuracy of the master curves obtained from the discrete and continuous spectrum models is evaluated.

Materials
The type of asphalt mixtures used in this study was AC-13 dense grading asphalt mixtures, as shown in Figure 1.The asphalt used No. 70 base asphalt produced in Tianjin.The coarse and fine aggregates and mineral powder come from limestone produced in Tongzhou.In this study, the fiber content was selected as 0%, 0.1%, 0.2% and 0.3% (the fiber content accounted for the mass fraction of the asphalt mixture), and for the convenience of the study, they were numbered as BF-0, BF-1, BF-2 and BF-3 ,respectively.The basalt fiber asphalt mixture is prepared by dry mixing method to ensure that the basalt fiber is fully dispersed in the asphalt mixtures.

The Time-Temperature Superposition Principle (TTSP)
The viscoelasticity of asphalt mixture has a strong dependence on time and temperature, and its viscoelastic behavior shows the equivalence of time and temperature, which is called TTSP [19].
When TTSP is applied, the test curves at different temperatures can be moved to the target temperature curve, and the moving distance is called the shift factor   where n is the total data points of the test,

Construction of the master curves of
can also be described by the GSM.
in equation (11), the master curve model of Selecting 20°C as the reference temperature, the model parameters of where n is the total data points of the test,

Determining the discrete spectrum models
As the most commonly used discrete spectrum, the Prony series model is widely used to characterize the linear viscoelastic behavior of asphalt mixtures.The discrete spectrum composed of the coefficients of the Prony series contains all the viscoelastic information.The discrete spectrum can organically combine the viscoelastic parameters in different domains, and then realize the transformation of the master curve of the viscoelastic parameters.

Determining the discrete relaxation spectrum
According to the Boltzmann superposition principle, the integral constitutive relation of the From the GMM and the GKM, the Prony series expression of   Etand   Jt can be deduced as follows [24]: where e E is the equilibrium modulus; i E is the relaxation strength; i  is the relaxation time; g J is the glassy compliance; j J is the retardation strength; j  is the retardation time; and t is the test time.
Using the Carson transform, the following relationship can be obtained from equations ( 14)-( 16) [25]: where   Es is the operator modulus,   Jsis the operator compliance; s is the Lapace transform operator.
Since the complex function pair of viscoelastic material is excited by a sine wave.Therefore, the relationship between it and the operator function in the frequency domain is where According to ( 18) and ( 19), it can be obtained that   * E  and   * J  are reciprocals of each other in the frequency domain [26].
According to Eq.( 15)-(20), the expressions of  can be obtained by using Carson transform as[27]: From Eq. ( 21) and ( 22), it can be known that under the specified relaxation time, Jtare brought into this equation, and the discrete retardation spectrum can be obtained.Substituting Eq. ( 15) and ( 16) into equation ( 14), we get Where    denotes the Dirac delta function.
Further operation of Eq. ( 25) can be converted into A or ; k t is the upper limit of the integral of formula ( 14).From Eq. ( 26)-( 28), it can be seen that when the discrete relaxation spectrum and When the time interval of discrete spectrum is close to infinitely small, it will evolve into continuous spectrum.The conversion between the models of the viscoelastic parameters and the continuous spectrum can be realized by using the integral transformation theory.According to the internal relationship between the storage modulus model (ie the GSM) and the continuous relaxation The integral relationship between   Et and the According to the Eq. ( 18), the integral expression of the   Separating the real and imaginary parts of Eq.( 30), the integral relation of the with respect to the time spectrum is obtained as Applying Stieltjes and its inverse transform to Eq. (31), The model of , use Euler's formula again to further simplify the formula, and finally obtain the relationship between

 
H  and the parameters of the model of

Determining the Continuous retardation Spectrum
As shown in Eq. ( 40), the retardation spectrum function can be derived from the known relaxation spectrum function [28].In order to ensure a one-to-one correspondence between the retardation time and the relaxation time during calculation, the retardation time is also represented by .

 
 are shown in Figures 2 and 3.At high frequency, the dynamic modulus of BF-2 is the smallest, the dynamic modulus of BF-0 is the largest, and the dynamic modulus of BF-0 and BF-3 are both smaller than BF-0.At low frequency, the dynamic modulusof BF-1, BF-2 and BF-3 are all larger than BF-0, and the dynamic modulus of the three groups of basalt fiber asphalt mixtures have little difference.According to TTPS, it shows that the basalt fiber asphalt has good mixed deformation ability and strong toughness at low-temperature, and can better resist the action of external loads and disperse the stress.At high frequency, the storage modulus first decreases and then increases with the increase of basalt fiber content, and the loss modulus decreases with the increase of basalt fiber content.At low frequency, the storage modulus also increases with the increase of basalt fiber content, and the loss modulus first decreases and then increases with the increase of basalt fiber content.It shows that at low temperature, adding basalt fiber can reduce the energy stored and lost in the asphalt mixture, and correspondingly reduce its low-temperature elastic capacity and low-temperature viscous deformation.

Construction of master curves of  
Et and   Jtfrom discrete spectrum

Determining the discrete relaxation spectrum
A discrete relaxation spectrum containing all viscoelastic information was obtained using or the collocation method.To reduce the fluctuation of the curve, the relaxation time are taken at intervals of 1 on the logarithmic time axis.Using Equation (42) to minimize the target error, the calculated Prony series results of the relaxation modulus are shown in Table 4.
Where n is the number of data points,  The discrete retardation spectrum models can be derived using equations ( 26)-( 28).This process generally requires three steps, namely determining the retardation time and k t ,calculating the retardation intensity and determining the glassy compliance.First, the discrete delay time can be calculated by Equation ( 43).The roots of Equation ( 43) can be quickly obtained by using the graphical root-finding method.Taking the absolute value of

 
Es as the ordinate and -1/s as the abscissa, calculate 1500 data points and draw a graph.The distribution curve of   Esand -1/s is shown in Figure 8.The abscissa of the minimum value in the figure is the discrete delay time.

 
Where N is the total number of data points, N=30.
Finally, according to the initial-value and final-value theorems of Laplace transform, the glassy compliance is determined using Eq. ( 45) Table 5 shows the coefficients of the creep compliance Prony series obtained by the transformation of the relaxation spectrum of the four asphalt mixtures.The discrete relaxation and retardation spectrum are drawn according to Table 4 and Table 5, as shown in Figure 9 and 10.Both discrete spectrum have obvious asymmetry.And with the increase of basalt fiber content, the peak relaxation strength showed a trend of decreasing first, while the discrete retardation spectrum did not have a specific change rule.
10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 The relaxation modulus reaches its maximum value when the loading time is short.When the loading time is very long, the relaxation modulus reaches the minimum value.And the relaxation modulus was found to vary greatly from 5 10  to 5 10 .This is because, the internal stress of the asphalt mixture is accumulated for a short loading time.With the extension of the loading time, the internal stress dissipates rapidly, and a small amount of stress remains after a long time of relaxation.At the same time, in a short loading time, the relaxation modulus decreased with the increase of basalt fiber content and then increased, and the relaxation modulus was the smallest when the basalt fiber content was 0.2%.It shows that basalt fiber can improve the low-temperature relaxation ability and the thermal shrinkage cracking resistance of asphalt mixtures, and reduce the accumulation of temperature stress of asphalt mixture.This is because basalt fiber improves the stress diffusion ability of the asphalt mixture, thereby improving the low-temperature crack resistance.In a long loading time, the relaxation modulus first increased and then decreased with the increase of basalt fiber content, and the relaxation modulus reached the maximum when the basalt fiber content was 0.2%.
When the loading time is short, the creep compliance tends to the minimum value; when the loading time is very long, the creep compliance tends to the maximum value, and the creep compliance changes the largest from 5 10  to 5 10 s .This phenomenon can be attributed to the softening of asphalt at high-temperature.As a whole, the asphalt mixture is composed of additives such as coarse and fine aggregates, asphalt, and mineral powder.With the increase of temperature or loading time, the viscosity of asphalt increases.The effect of the knot is weakened, which leads to a reduction in the overall resistance to deformation of the asphalt.At the same time, in the short loading time, the creep compliance first increases and then decreases with the basalt fiber content, and the creep compliance of BF-2 is the largest.For a longer loading time, the creep compliance decreases and then increases with the increase of the basalt fiber content, and the creep compliance of BF-2 is the smallest.This shows that basalt fiber can improve the deformation ability and resist creep deformation at high temperature of the asphalt mixture.And correspondingly improve its antirutting ability.Moreover, when the basalt fiber content is 0.2%, the effect of improving the deformation resistance of the asphalt mixture is the best.

Construction master curves of   Et and  
Jt from Continuous spectrum

Determining the continuous relaxation Spectrum
The model parameters of 3 are put into Eq.(39), and the continuous relaxation spectrum model of four kinds of asphalt mixture can be obtained.Taking the relaxation time as the abscissa and

 
H  as the ordinate, the continuous relaxation spectrum when the reference temperature is 20°C is obtained, as shown in Figure 13.The relaxation spectrum is asymmetric.And the peak of the continuous spectrum is the same as that obtained by the discrete relaxation time spectrum.The parametric instruments used were derived from master curves of


. The constructed continuous relaxation spectrum therefore contains all the linear viscoelastic information.

   
The numerical expression of  

 
Et of the four asphalt mixtures were calculated according to Eq.( 48), and construct the master curves，as shown in Fig. 15.The master curves of

 
Et obtained from continuous and discrete relaxation spectrum have the same trend of variation with loading time.Moreover, at short and long loading times, the master curves of   Et with the basalt fiber content has the same trend, also.In Eq.( 50), there is still one unknown Jtobtained from the continuous and discrete retardation spectrum has the same trend with loading time.Moreover, the creep compliance changes with the basalt fiber content in the same trend in short loading time, also.However, the creep compliance of BF-3 is larger than that of Bf-2 in a very short loading time.And it is larger than that of BF-0 in a very long loading time.
These reasons can be seen from the continuous and the discrete spectrum.21) and ( 22), E and E of each of the four asphalt mixtures at the reduced frequency can be directly calculated.

When calculating
E and E from

 
H  , Eq.( 31) and (32) are calculated using the chained trapezoidal rule, the integral interval is selected as   .This shows that the discrete and continuous relaxation spectrum models can not only construct the relaxation modulus master curve with high accuracy, but also the obtained relaxation modulus master curve has little difference in accuracy.
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EHL
Et, and creep compliance   Jt are commonly used to describe the viscoelastic behavior of asphalt mixtures[2].Different viscoelastic parameters characterize the different viscoelastic response.Among them,   * E  ,   E   and   E   are all functions with respect to frequency and describe the strength properties, purely elastic behavior and purely viscous behavior of the asphalt mixtures, respectively[3].  Etand   Jtare all functions of time and describe the stress relaxation and deformation capacity of the asphalt mixture, respectively[4 is an important parameter for asphalt pavement design, and   Etand   Jt can be used to predict the temperature stress and deformation of asphalt pavements[5].However, the cost of obtaining accurate   Et and   Jt is significant due to factors such as test instrumentation, test time and test operability.In contrast, complex modulus tests are simpler to operate and the tests are more easily controlled[6].According to the theory of linear viscoelasticity, these response functions are interrelated, and the conversion of viscoelastic parameters to each other can be realized through mathematical theories[7].Therefore, it is practical to constant the master curves of   Et and   Jt from the linear viscoelasticity theory and the complex modulus test results[8] .The Boltzmann superposition principle provides a theoretical basis for the conversion between the models of the viscoelastic parameters[9].The models of   Etand   Jtare usually determined tests using discrete and continuous models[10][11].The most commonly used discrete models of   Etand   Jtis the Prony series model.Using this model to obtain the model of   Et and   Jt mainly includes four steps.First, the master curves of storage compliance   the purpose of pre-smoothing the experimental data.Second, the Prony series model of the generalized Kelvin model(GKM)[12].Third, the coefficients in the Prony series were determined by fitting the models of the discrete model of   Jtis determined according to the transformation relationship between Et ; and ,the discrete model of   Jt is determined according to the transformation relationship between steps are required to determine   Et and   Jt using the continuous spectrum models[13][15].First, the models of fitted.Second, an expression for the continuous relaxation spectrum  H  with respect to the model of  .And the expression of the continuous retardation spectrum   L  with respect to the model of    .Third, the continuous relaxation and retardation spectrum models are determined by substituting the models of into this expressions, respectively.Fourth, the equilibrium modulus is determined from the model of continuous relaxation spectrum.And the glassy compliance is determined from the model of the continuous retardation spectrum.Finally, the numerical models of   Et and   Jt are established by using the numerical integral formula.There are two main drawbacks of using the above discrete and continuous spectrum models for the master curve conversion of viscoelastic parameters.First, the conversion process ignores the information provided by loss modulus and loss compliance, and the constructed the master curves of   Etand  Jt do not contain all the viscoelastic information.Secondly, fitting the models of number of fits and reduces the accuracy of the converted master curves.Basalt fiber, as a new road material, has received a lot of attention in recent years.Previous studies on the viscoelastic behavior of basalt fiber asphalt mixtures have more often used dynamic modulus and phase angle to evaluate their viscoelastic behavior, and rarely studied their viscoelastic properties comprehensively[16][17].Moreover, many studies characterizing the viscoelasticity of basalt fiber asphalt mixtures only plot the dynamic modulus master curve and ignore the phase angle master curve; or choose two unrelated models to plot the dynamic modulus and phase angle master curves separately.The model functions thus obtained do not satisfy the K-K relationship or do not conform to the linear viscoelasticity theory.The research methods for the relaxation and deformation properties of asphalt mixtures are also limited to static tests that differ significantly from the actual road stresses in service.In view of the above problems, to evaluate the difference of the master curve obtained by the transformation of discrete and continuous models, and to achieve the purpose of comprehensively characterizing the viscoelastic behavior of basalt fiber asphalt mixtures.In this study, the approximate Kramers-Kronig(K-K)relationship and the generalized Sigmoidal model(GSM) are applied to construct the master curves of dynamic modulus, phase angle, storage modulus and loss modulus.Then, the master curves of   Et containing all viscoelastic information are constructed from the master curves of continuous spectrum models, respectively.Next, the master curves of  Jt is derived from the integral relationship between the

t.,
In this paper, when ‫؟‬ ‫؟‬ constructing the master curves the Williams-Landel-Ferr (WFL) equation shown in Eq.(5) is chosen to calculate , because its parameters have clear physical meaning and the calculation results are more accurateCC is the material constant; T is the test temperature; and r T is the reference temperature.3.3 Construction of master curves in frequency domain 3.3.1 Construction of the master curves of   * E  and    In order to ensure that the constructed the master curves of   * E  and    both conform to linear viscoelasticity and allow the master curves to be unsymmetrical about the viewpoint, the (GSM)and the approximate K-K relationship are used, as shown in Eq (6)-(8)[1][21].

E
 ,  is the difference between the maximum and minimum values of   * E  ;  、  and  are the shape coefficients.Selecting 20°C as the reference temperature, the model parameters of   * E  and    are simultaneously solved by minimizing the parameter error.The error function is


is the phase The logarithmic value of    .
at different frequencies.The obtained parameters are put into equation (15) to get the master curve of   Jt. 3.4.2Determining the discrete retardation spectrum models   Et and   Jt satisfy the integral relationship shown in Eq. (14), and the Prony series expressions of   Et and  

t
are set, the discrete retardation spectrum models can be obtained by solving the inhomogeneous linear equation system, and then   Et can be obtained.3.5 Determining the continuous spectrum 3.5.1 Determining the continuous relaxation spectrum models Using Euler's equation, the complex exponential function of Eq. (34) is converted into a trigonometric function, and the expression of   H  is obtained by further solving Linear viscoelasticity characterization of basalt fiber asphalt mixture 4.1.1Master curves of   * E  and    Use Excel's Solver function to calculate equation (9) to obtain the model parameters of   * E  and    .The fitting results are shown in

Fig. 4
Fig. 4 Comparison the master curves of

Fig. 6
Fig.6 Master curves of is the calculated value of the loss modulus Prony series.

.
Solving the inhomogeneous system of linear equations requires pn  because of ij   , and setting retardation strength is determined by solving the error function equation (26).

4.3. 2
Constructed the continuous retardation spectrum from the continuous relaxation spectrum When deriving the continuous retardation spectrum from Eq.(40), the integral term   Z  and the equilibrium modulus e E need to be calculated.Use the chained trapezoidal rule to solve the integral term   Z  .The integration interval of is selected as , the number of sub-intervals is 14000, and the length of the sub- Fig. 13 Continuous relaxation spectrum Fig. 14 Continuous retardation spectrum of asphalt mixtures of asphalt mixtures 4.3.3Constructed master curves of   Et and   Jtbased on continuous spectrum Use the chained trapezoidal rule to solve Eq. (29), the integration interval is selected as 40 30 10 ,10    , and the subinterval length is selected as log

gJ
Fig.16.The master curves of  

4
Fig. 15 Master curve of   Et from continuous Fig.17 Relative errors of   Et Fig. 19.Comparison of calculated E and E with the master curves of

4.5. 1
Accuracy verification of the master curves of   Jt According to Eq.(20)   * E  and   * J  relationship, it is deduced that the storage compliance (58)  and (59) are discretized by the chained trapezoidal rule, the integral interval is

R
Figure20.The master curves of written in non-logarithmic form and replacing with Table 1, and the master curves of * E  and

Table 1
Models parameters of * E  and  

Table 3
Prony series coefficients of